Question

Find the equation of the ellipse referred to its major and minor axes as the coordinate axes, $X,Y-$ respectively with latus rectum of length $4$, and distance between foci $4\sqrt{2}$.

Answer 1

lenght of latus rectum of ellipse=$\frac{2b^2}{a}=4$

distance between focii=$2ae=2a\sqrt{1-\frac{b^2}{a^2}}=4\sqrt{2}$

from 1st equation;$b^2=2a$

$\Rightarrow 2a\sqrt{1-\frac{2}{a}}=4\sqrt{2}$

$\Rightarrow 4a^2(1-\frac{2}{a})=32$

$\Rightarrow a^2-2a-8=0$

$\Rightarrow (a-4)(a+2)=0$

$\Rightarrow a=4$

$\Rightarrow b=2\sqrt{2}$

Equation of ellispe = $\frac{x^2}{16}+\frac{y^2}{8}=1$